Morley's Theorem

The locus of the incenters of the cardioids that touch 3 lines are in general position is a union of three sets of three parallel lines forming angles of 60° (and, naturally, 120°.) These are the axes of the 3-line. Naturally, the axes form equilateral triangles - 27 of them. An important observation is that the vertices of these triangles - the points where two axes meet - have the property that the corresponding cardioid touches one of the given lines (of the 3-line) twice.